Question
Use spherical coordinates and evaluate V1-r2 VA-:? 4? K $" rdzdydx Vs?+y?
Use spherical coordinates and evaluate V1-r2 VA-:? 4? K $" rdzdydx Vs?+y?
Answers
Use spherical coordinates.
Evaluate $\iiint_{E}\left(x^{2}+y^{2}\right) d V,$ where $E$ lies between the spheres
$x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}+z^{2}=9$
Were given an integral. And we're asking you spherical coordinates to evaluate it. Sorry mm. This is the triple integral. Excuse me. Over the region E. He ordered cafe con Leche, sweating out the order coffee militants. That's what we use a question. You and your boys have been hanging out. Yeah. Of X squared plus Y squared D V. Where E is the region that lies between the spheres are oh, first of all, this is steel, X squared plus Y squared plus C scrape was four. Use the voice over at sausage party, homework in X squared plus Y squared plus B squared equals nine. Yes, he is played. He played Andy's little sister. Mhm. Green. To make the change to spherical coordinates first. Right E in spherical coordinates. So because we live between these two spheres, we know that rho squared is going to lie between four and nine. So that row lies between the two radio guys, two and three of the spheres. As for phi and theta wealthy. That ranges from zero to pi still and five from zero to pi sausage. So some you hate now we can use spherical coordinates to write are integral as the integral from 0 to 2 pi integral from zero to pi Integral from 2 to 3 of our function. But now in spherical coordinates X squared plus y squared becomes rho squared sine squared phi. Okay. And then D V becomes rho squared sine phi Hero D five d. Theta sauce label. Like I feel like We can rewrite this using Fujianese theorem as a proactive integral and we get the integral from 0 to 2 pi the theater times the integral from zero to pi of sine cubed. Fi defy Times the integral from 2 to 3 of Road to the 4th. Dear. Oh, it was really a chicken nugget. Also, chinese in Wisconsin. Appleton Taking anti derivatives. We get data from data equals 0-2 pi. The anti derivative of sine cubed. If I this is a negative co sign fi these are plus co sine cubed. If I different sauce cover Syria Children's Over three. I mean felt like I was going to destroy got some real California voice from phi equals zero to pi times. Uh Yeah it's actually named Road to the 5th over five from row equals 2 to 3 shows to the next month. Your mom and your so plugging in values We get two pi times. Use italian the fuck mama. The ranking male men or thirds. Mm The ceremony. Thank you Germany. Yes. Do you? How many people fuck by the really great pleasure kissing you directly. And so yeah. Mm. Times. And this is 3 to the 5th -2 to the 5th Over five. Mhm. And solving this is equal to 1688 pie over 15. Excuse me get married. Sure candidate.
So in this sort of problems, we want to convert the cylindrical coordinates to spherical coordinates. So for example, if we're given something like four, I have worked for um zero. What might be easiest for us to do is to convert this from polar to rectangular or from cylindrical to rectangular and then from rectangular to spherical. So we know that the way to do this would be by considering the fact that Feta is Pi over four, Z is going to be zero. So when this is in the X Y Z coordinate system, we know that it's going to be Route 2/2 times for so it's going to be we're going to have are different polar or rectangular coordinates. So if we consider this to be what we're looking for, we see that going from cylindrical to rectangular, we're going to have X is equal to our coastline data. So it's going to be for October two. So two route to and then this will be to route to so the zero joe. But so I can rectangular now. And then when we go to row, for example, that's going to be the square root of expert plus Y script, Z squared, that's gonna be um 8-plus 8, which is 16 squared of that is going to be four. And then we're going to get our angle, we know feta is going to be the same as data obviously. So it's gonna be pi over four. And then lastly we want to find fee which we can do using the fact that fee is equal to um the inverse co sign of Z over X squared plus Y squared, posey scored. And that'll give us pipe rigid. So it can be the process through which we go and it's going to involve that's converting from cylindrical to rectangular and rectangular, spherical. So we're using rectangular coordinates as the middleman. It might seem a little bit inefficient, but it will actually save you some time in the long run because it's a little complicated to go from cylindrical and spherical.
So in this sort of problems, we want to convert the cylindrical coordinates to spherical coordinates. So for example, if we're given something like four, I have worked for um zero. What might be easiest for us to do is to convert this from polar to rectangular or from cylindrical to rectangular and then from rectangular to spherical. So we know that the way to do this would be by considering the fact that Feta is Pi over four, Z is going to be zero. So when this is in the X Y Z coordinate system, we know that it's going to be Route 2/2 times for so it's going to be we're going to have are different polar or rectangular coordinates. So if we consider this to be what we're looking for, we see that going from cylindrical to rectangular, we're going to have X is equal to our coastline data. So it's going to be for October two. So two route to and then this will be to route to so the zero joe. But so I can rectangular now. And then when we go to row, for example, that's going to be the square root of expert plus Y script, Z squared, that's gonna be um 8-plus 8, which is 16 squared of that is going to be four. And then we're going to get our angle, we know feta is going to be the same as data obviously. So it's gonna be pi over four. And then lastly we want to find fee which we can do using the fact that fee is equal to um the inverse co sign of Z over X squared plus Y squared, posey scored. And that'll give us pipe rigid. So it can be the process through which we go and it's going to involve that's converting from cylindrical to rectangular and rectangular, spherical. So we're using rectangular coordinates as the middleman. It might seem a little bit inefficient, but it will actually save you some time in the long run because it's a little complicated to go from cylindrical and spherical.
Okay. In this problem, we want to convert the point given spherical coordinates to rectangular coordinates instead were given row satya sai equals 9.4 high. This question is challenging understanding of how to convert convert between coordinate systems in the same three D. Space, namely its testing an understanding of a kingly coordinate systems X, Y. Z. And spherical rotate aside. To convert between these chords systems. We need to know the proper conversion factors. These are as follows, X equals road signs because data Y equals rothstein decide data Z equals rose to speed. The inverse of these three formulas are given below. So from here we can solve as follows, X equals nine signed pie coastline pi over four or nine times zero times route to over two equals zero. Why is 97 pricing power four. Again, some sign pi zero. That's zero. Finally Z is nine coastline pi. Cosine pi is negative one. So this is nine times one is negative nine. That's this point. As long as the access and the negative and we have 90.0.0, negative nine.